= Total branch length of a neutral coalescent
{title2=$L=\sum_{j=2}^n jT_j$}
In coalescent units, independent waiting times satisfy $T_j\sim\operatorname{Exp}(j(j-1)/2)$, and total branch length is $L=\sum_{j=2}^n jT_j$. Therefore $EL=2\sum_{i=1}^{n-1}1/i$. The variables $jT_j/2$ are independent exponentials with rates $j-1$, so $W=L/2$ has the same distribution as the maximum of $n-1$ independent unit exponentials, with density $(n-1)e^{-w}(1-e^{-w})^{n-2}$. The last identity follows either from its Laplace transform or from the independent spacings of exponential order statistics.
Back to article page
